Efficient Pricing of Low Volatility Path Dependent Options

Abstract

Asian options are generally priced using arithmetic or geometric averages of the underlying stock. However, these methods are not suitable when stock’s volatilities are very low. The motivation to develop derivative prices based on averaging the underlying asset stems from the robust features associated with Asian options which suggest that they are more suited to African markets where prices can be dormant for long periods resulting in low volatilities in stock prices. We propose the use of the modal average as the measure of the underlying stock price when stocks have low volatilities instead of the more popular arithmetic and geometric averages. In particular, the stock price is assumed to follow Geometric Brownian Motion and using the concept of maximum of a function, a model for the modal average of the underlying stock is derived. A process of obtaining the price of a call option is subsequently developed. Theoretically, we prove further that for very low volatilities the modal average model is a better estimator of the expected average of the stock price and consequently produces cheaper option prices than geometric and arithmetic average models. Using data from the Ghana Stock Exchange and the Nasdaq, the proposed model is used to price options sold on selected stocks on the exchange. The numerical results consistently show that for underlying stocks with volatility less than 3%, the modal average model provides cheaper call options than the arithmetic or geometric averages pricing models.